Rend. Sem. Mat. Univ. Pol. Torino
Vol. 58, 2 (2000)
Geom., Cont. and Micros., II
B. Svendsen∗
NON-LOCAL CONTINUUM THERMODYNAMIC
EXTENSIONS OF CRYSTAL PLASTICITY TO INCLUDE
THE EFFECTS OF GEOMETRICALLY-NECESSARY
DISLOCATIONS ON THE MATERIAL BEHAVIOUR
Abstract. The purpose of this work is the formulation of constitutive models
for the inelastic material behaviour of single crystals and polycrystals in which
geometrically-necessary dislocations (GNDs) may develop and influence this behaviour. To this end, we focus on the dependence of the development of such dislocations on the inhomogeneity of the inelastic deformation in the material. More
precisely, in the crystal plasticity context, this is a relation between the density of
GNDs and the inhomogeneity of inelastic deformation in glide systems. In this
work, two models for GND density and its evolution, i.e., a glide-system-based
model, and a continuum model, are formulated and investigated. As it turns out,
the former of these is consistent with the original two-dimensional GND model of
Ashby (1970), and the latter with the more recent model of Dai and Parks (1997).
Since both models involve a dependence of the inelastic state of a material point on
the (history of the) inhomogeneity of the glide-system inelastic deformation, their
incorporation into crystal plasticity modeling necessarily implies a corresponding non-local generalization of this modeling. As it turns out, a natural quantity
on which to base such a non-local continuum thermodynamic generalization, i.e.,
in the context of crystal plasticity, is the glide-system (scalar) slip deformation.
In particular, this is accomplished here by treating each such slip deformation as
either (1), a generalized “gradient” internal variable, or (2), as a scalar internal
degree-of-freedom. Both of these approaches yield a corresponding generalized
Ginzburg-Landau- or Cahn-Allen-type field relation for this scalar deformation
determined in part by the dependence of the free energy on the dislocation state in
the material. In the last part of the work, attention is focused on specific models for
the free energy and its dependence on this state. After summarising and briefly discussing the initial-boundary-value problem resulting from the current approach as
well as its algorithmic form suitable for numerical implementation, the work ends
with a discussion of additional aspects of the formulation, and in particular the
connection of the approach to GND modeling taken here with other approaches.
∗ I thank Paolo Cermelli for helpful discussions and for drawing my attention to his work and that of
Morton Gurtin on gradient plasticity and GNDs.
207
208
B. Svendsen
1. Introduction
Standard micromechanical modeling of the inelastic material behaviour of metallic single crystals and polycrystals (e.g., Hill and Rice, 1972; Asaro, 1983; Cuitiño and Ortiz, 1992) is commonly based on the premise that resistance to glide is due mainly to the random trapping of
mobile dislocations during locally homogeneous deformation. Such trapped dislocation are commonly referred to as statistically-stored dislocations (SSDs), and act as obstacles to further dislocation motion, resulting in hardening. As anticipated in the work of Nye (1953) and Kröner
(1960), and discussed by Ashby (1970), an additional contribution to the density of immobile
dislocations and so to hardening can arise when the continuum lengthscale (e.g., grain size) approaches that of the dominant microstructural features (e.g., mean spacing between precipitates
relative to the precipitate size, or mean spacing between glide planes). Indeed, in this case,
the resulting deformation incompatibility between, e.g., “hard” inclusions and a “soft” matrix,
is accomodated by the development of so-called geometrically-necessary dislocations (GNDs).
Experimentally-observed effects in a large class of materials such as increasing material hardening with decreasing (grain) size (i.e., the Hall-Petch effect) are commonly associated with the
development of such GNDs.
These and other experimental results have motivated a number of workers over the last
few years to formulate various extensions (e.g., based on strain-gradients: Fleck and Hutchinson, 1993, 1997) to existing local models for phenomenological plasticity, some of which have
been applied to crystal plasticity (e.g., the strain-gradient-based approach: Shu and Fleck, 1999;
Cosserat-based approach: Forest et al., 1997) as well. Various recent efforts in this direction
based on dislocation concepts, and in particular on the idea of Nye (1953) that the incompatibility of local inelastic deformation represents a continuum measure of dislocation density (see
also Kröner, 1960; Mura, 1987), include Steinmann (1996), Dai and Parks (1997), Shizawa and
Zbib (1999), Menzel and Steinmann (2000), Acharya and Bassani (2000), and most recently
Cermelli and Gurtin (2001). In addition, the recent work of Ortiz and Repetto (1999) and Ortiz
et al. (2000) on dislocation substructures in ductile single crystals demonstrates the fundamental
connection between the incompatibility of the local inelastic deformation and the lengthscale of
dislocation microstructures in FCC single crystals. In particular, the approaches of Dai and Parks
(1997), Shizawa and Zbib (1999), and Archaya and Bassani (2000) are geared solely to the modeling of additional hardening due to GNDs and involve no additional field relations or boundary
conditions. For example, the approach of Dai and Parks (1997) was used by Busso et al. (2000)
to model additional hardening in two-phase nickel superalloys, and that of Archaya and Bassani
(2000) by Archaya and Beaudoni (2000) to model grain-size effects in FCC and BCC polycrystals up to moderate strains. Except for the works of Acharya and Bassani (2000) and Cermelli
and Gurtin (2001), which are restricted to kinematics, all of these presume directly or indirectly
a particular dependence of the (free) energy and/or other dependent constitutive quantities (e.g.,
yield stress) on the gradients of inelastic state variables, and in particular on that of the local
inelastic deformation, i.e., that determine its incompatibility. Yet more general formulations of
crystal plasticity involving a (general) dependence of the free energy on the gradient of the local
inelastic deformation can be found in, e.g., Naghdi and Srinivasa (1993, 1994), Le and Stumpf
(1996), or in Gurtin (2000).
From the constitutive point of view, such experimental and modeling work clearly demonstrates the need to account for the dependence of the constitutive relations, and so material
behaviour, on the inhomogeneity or “non-locality” of the internal fields as expressed by their
gradients. In the phenomenological or continuum field context, such non-locality of the material
behaviour is, or can be, accounted for in a number of existing approaches (e.g., Maugin, 1980;
Non-local continuum thermodynamic
209
Capriz, 1989; Maugin, 1990; Fried and Gurtin, 1993, 1994; Gurtin, 1995; Fried, 1996; Valanis,
1996, 1998) for broad classes of materials. It is not the purpose of the current work to compare
and contrast any of these with each other in detail (in this regard, see, e.g., Maugin and Muschik,
1994; Svendsen, 1999); rather, we wish to apply two of them to formulate continuum thermodynamic models for crystal plasticity in which gradients of the inelastic fields in question influence
the material behaviour. To this end, we must first identify the relevant internal fields. On the
basis of the standard crystal plasticity constitutive relation for the local inelastic deformation P ,
a natural choice for the principal inelastic fields of the formulation is the set of glide-system
deformations. In contrast, Le and Stumpf (1996) worked in their variational formulation directly
with P , and Gurtin (2000) in his formulation based on configurational forces with the set of
glide-system slip rates. In both of these works, a principal result takes the form of an extended
or generalized Euler-Lagrange-, Ginzburg-Landau- or Cahn-Allen-type field relation for the respective principal inelastic fields. Generalized forms of such field relations for the glide-system
deformations are obtained in the current work by modeling them in two ways. In the simplest
approach, these are modeled as “generalized” internal variables (GIVs) via a generalization of
the approach of Maugin (1990) to the modeling of the entropy flux. Alternatively, and more generally, these are modeled here as internal degrees-of-freedom (DOFs) via the approach of Capriz
(1989) in the extended form discussed by Svendsen (2001a). In addition, as shown here, these
formulations are general enough to incorporate in particular a number of models for GNDs (e.g.,
Ashby, 1970; Dai and Parks, 1997) and so provide a thermodynamic framework for extended
non-local crystal plasticity modeling including the effects of GNDs on the material behaviour.
After some mathematical preliminaries (§2), the paper begins (§3) with a brief discussion
and formulation of basic kinematic and constitutive issues and relations relevant to the continuum
thermodynamic approach to crystal plasticity taken in this work. In particular, as mentioned
above, the standard constitutive form for P in crystal plasticity determines the glide-system
slip deformations (“slips”) as principal constitutive unknowns here. Having then established the
corresponding constitutive class for crystal plasticity, we turn next to the thermodynamic field
formulation and analysis (§§4-5), depending on whether the glide-system slips are modeled as
generalized internal variables (GIVs) (§4), or as internal degrees-of-freedom (DOFs) (§5). Next,
attention is turned to the formulation of two (constitutive) classes of GND models (§6), yielding
in particular expressions for the glide-system effective (surface) density of GNDs. The first class
of such models is based on the incompatibility of glide-system local deformation. To this class
belong for example the original model of Ashby (1970) and the recent dislocation density tensor
of Shizawa and Zbib (1999). The second is based on the incompatibility of P and is consistent
with the model of Dai and Parks (1997). With such models in hand, the possible dependence of
the free energy on quantities characterising the dislocation state of the material (e.g., dislocation
densities) and the corresponding consequences for the formulation are investigated (§7). Beyond
the GND models formulated here, examples are also given of existing SSD models which can be
incorporated into models for the free energy, and so into the current approach. After discussing
simplifications arising in the formulation for the case of small deformation (§8), as well as the
corresponding algorithmic form, the paper ends (§9) with a discussion of additional general
aspects of the current approach and a comparison with other related work.
2. Mathematical preliminaries
If W and Z represent two finite-dimensional linear spaces, let Lin(W, Z) represent the set of
all linear mappings from W to Z. If W and Z are inner product spaces, the inner products
on W and Z induce the transpose T ∈ Lin(Z, W ) of any ∈ Lin(W, Z), as well as the inner
210
B. Svendsen
product · : = trW ( T ) = tr Z ( T ) on Lin(W, Z) for all , ∈ Lin(W, Z). The main
linear space of interest in this work is of course three-dimensional Euclidean vector space V . Let
Lin(V , V ) represent the set of all linear mappings of V into itself (i.e., second-order Euclidean
tensors). Elements of V and Lin(V , V ), or mappings taking values in these spaces, are denoted
here as usual by bold-face, lower-case , . . . and upper-case , . . ., italic letters, respectively.
In particular, ∈ Lin(V , V ) represents the second-order identity tensor. As usual, the tensor
product ⊗ of any two , ∈ V can be interpreted as an element ⊗ ∈ Lin(V , V ) of
Lin(V , V ) via ( ⊗ ) : = ( · ) for all , , ∈ V . Let sym( ) : = 21 ( + T ) and
skw( ) : = 21 ( − T ) represent the symmetric and skew-symmetric parts, respectively, of
any ∈ Lin(V , V ). The axial vector axi( ) ∈ V of any skew tensor ∈ Lin(V , V ) is defined
by axi( ) × : = . Let , , ∈ V be constant vectors in what follows.
Turning next to field relations, the definition
(1)
curl : = 2 axi(skw(∇ ))
for the curl of a differentiable Euclidean vector field is employed in this work, ∇ being the
standard Euclidean gradient operator. In particular, (1) and the basic result
(2)
∇( f ) = ⊗ ∇f + f (∇ )
for all differentiable functions f and vector fields yield the identity
(3)
curl ( f ) = ∇f × + f (curl )
.
In addition, (1) yields the identity
(4)
curl · × = ∇ · − ∇ · for curl in terms of the directional derivative
(5)
∇ : = (∇ ) of in the direction ∈ V . Turning next to second-order tensor fields, we work here with the
definition∗
(6)
(curl )T : = curl (
T )
for the curl of a differentiable second-order Euclidean tensor field as a second-order tensor
field. From (3) and (6) follows in particular the identity
(7)
curl ( f ) = ( × ∇f ) + f (curl )
for all differentiable f and , where ( × ) : = × . Note that ( × )T = × with
( × ) : = × . Likewise, (1) and (6) yield the identity
(8)
(curl )( × ) : = (∇ ) − (∇ ) for curl in terms of the directional derivative
∇ (∇ ) ∗ This is of course a matter of convention. Indeed, in contrast to (6), Cermelli and Gurtin (2001) define
(curl ) : = curl ( T ).
211
Non-local continuum thermodynamic
of in the direction ∈ V . Here, ∇ represents a third-order Euclidean tensor field. Let
be a differentiable invertible tensor field. From (8) and the identity
T ( × ) = det( ) ( × )
(9)
for any second-order tensor ∈ Lin(V , V ), we obtain
curl (
(10)
) (curl )
) = det(
−T
+ (curl
)
for the curl of the product of two second-order tensor fields. Here, curl represents the curl
operator induced by the Koszul connection ∇ induced in turn by the invertible tensor field
,
i.e.,
∇
(11)
: = (∇
)
−1
.
The corresponding curl operation then is defined in an analogous fashion to the standard form
(8) relative to ∇.
Third-order tensors such as ∇ are denoted in general in this work by , , . . . and interpreted as elements of either Lin(V , Lin(V , V )) or Lin(Lin(V , V ), V ). Note that any third-order
tensor induces one S defined by
(12)
(
S
) : = ( ) .
In particular, this induces the split
= sym ( ) + skw ( )
(13)
S
of any third-order tensor
S
into “symmetric”
sym ( ) : = 21 ( + )
S
(14)
S
and “skew-symmetric”
skwS ( ) : = 21 (
(15)
− )
S
parts. In addition, the latter of these induces the linear mapping
(16)
axiS : Lin(V , Lin(V , V )) −→ Lin(V , V )
7−→ = axi ( )
|
S
defined by
(17)
axiS ( )( × ) : = 2 (skwS ( ) ) = ( ) − ( ) .
With the help of (12)–(17), one obtains in particular the compact form
curl = axiS (∇ )
(18)
for the curl of a differentiable second-order tensor field as a function of its gradient ∇ from
(8). The transpose T ∈ Lin(Lin(V , V ), V ) of any third-order tensor ∈ Lin(V , Lin(V , V )) is
defined here via T · = · .
Finally, for notational simplicity, it proves advantageous to abuse notation in this work and
denote certain mappings and their values by the same symbol. Other notations and mathematical
concepts will be introduced as they arise in what follows.
212
B. Svendsen
3. Basic kinematic, constitutive and balance relations
Let B represent a material body, p ∈ B a material point of this body, and E Euclidean point space
with translation vector space V . A motion of the body with respect to E in some time interval
I ⊂ takes as usual the form
= ξ (t, p)
relating each p to its (current) time t ∈ I position
material velocity, and
(19)
∈ E in E. On this basis, ξ˙ represents the
κ
+
κ (t, p) : = (∇ ξ )(t, p) ∈ Lin (V , V )
the deformation gradient relative to the (global) reference placement κ of B into E. Here, we
are using the notation
∇ κ ξ : = κ ∗ (∇(κ ∗ ξ ))
for the gradient of ξ with respect to κ in terms of push-forward and pull-back, where (κ ∗ ξ )(t, κ )
: = ξ (t, κ −1 (κ )) for push-forward by κ , with κ = κ ( p), and similarly for κ ∗ . Like ξ , ξ˙ and
κ , all fields to follow are represented here as time-dependent fields on B . And analogous to
that of ξ in (19), the gradients of these fields are all defined relative to κ . More precisely, these
are defined at each p ∈ B relative to a corresponding local reference placement† at each p ∈ B ,
i.e., an equivalence class of global placements κ having the same gradient at p. Since κ and
the corresponding local reference placement at each p ∈ B is arbitrary here, and the dependence
of κ and the gradients of other fields, as well as that of the constitutive relations to follow, on
κ does not play a direct role in the formulation in this work, we suppress it in the notation for
simplicity.
In the case of phenomenological crystal plasticity, any material point p ∈ B is endowed
with a “microstructure” in the form of a set of n glide systems. The geometry and orientation
of each such glide system is described as usual by an orthonormal basis ( , , ) ( =
1, . . . , n ). Here, represents the direction of glide in the plane, the glide-plane normal, and
: = × the direction transverse to in the glide plane. Since we neglect in this work
the effects of any processes involving a change in or evolution of either the glide direction or the glide-system orientation (e.g., texture development), these referential unit vectors, and
so as well, are assumed constant with respect to the reference placement. With respect to the
glide-system geometry, then, the (local) deformation of each glide system takes the form of
a simple shear‡
(20)
= + γ ⊗ ,
γ being its magnitude in the direction of shear. For simplicity, we refer to each γ as the
(scalar) glide-system slip (deformation). The orthogonality of ( , , ) implies T = and = , as well as γ = · . In addition,
(21)
˙ = ⊗ γ˙ = : follows from (20). As such, the evolution of the glide-system deformation tensor
mined completely by that of the corresponding scalar slip γ .
† Refered to by Noll (1967) as local reference configuration of p ∈ B in E.
‡ As discussed in §6, like
P , and unlike , is in general not compatible.
is deter-
213
Non-local continuum thermodynamic
From a phenomenological point of view, the basic local inelastic deformation at each material point in the material body in question is represented by an invertible second-order tensor
field P on I × B . The evolution of P is given by the standard form
˙ =
P
P
(22)
P
in terms of the plastic velocity “gradient” P . The connection to crystal plasticity is then obtained via the constitutive assumption
Xm
Xm
P =
(23)
b
=
⊗ γ˙ =1
=1
for P via (21), where m ≤ n represents the set§ of active glide-systems, i.e., those for which
γ˙ 6= 0. Combining this last constitutive relation with (22) then yields the basic constitutive
expression
Xm
Xm
˙ =
(24)
P
P =
( ⊗ ) P γ˙ =1
for the evolution of
(25)
P.
=1
In turn, this basic constitutive relation implies that
Xm
˙
∇ P=
( ⊗ )(∇ P ) γ˙ + ( ⊗ )
=1
P
⊗ ∇˙
γ
for the evolution of ∇ P , and so that
(26)
˙
curl
P
=
Xm
=1 ( ⊗ )(curl
P ) γ˙
+ ⊗ (∇˙
γ ×
T
P
)
for the evolution of curl P via (7) and (8). On this basis, the evolution relation for P is linear
in the set γ˙ : = (γ˙1 , . . . , γ˙m ) of active glide-system slip rates. Similarly, the evolution relations
for ∇ P and curl P are linear in γ˙ and ∇˙
γ . Generalizing the case of curl P slightly, which
represents one such measure, the dislocation state in the material is modeled phenomenologically
in this work via a general inelastic state/dislocation measure α whose evolution is assumed to
depend quasi-linearly on γ˙ and ∇˙
γ , i.e.,
α̇ =
(27)
γ˙ + ∇˙
γ
in terms of the dependent constitutive quantities and . In particular, on the basis of (24), P
is considered here to be an element of α. In turn, the dependence of this evolution relation on ∇˙
γ
requires that we model the γ as time-dependent fields on B . As such, in the current thermomechanical context, the absolute temperature θ, the motion ξ , and the set γ of glide-system slips,
represent the principal time-dependent fields, P and α being determined constitutively by the
history of γ and ∇˙
γ via (24) and (27), respectively. On the basis of determinism, local action,
and short-term mechanical memory, then, the material behaviour of a given material point p ∈ B
is described by the general material frame-indifferent constitutive form
(28)
= (θ, , α, ∇θ, γ̇ , ∇ γ̇ , p)
for all dependent constitutive quantities (e.g., stress), where = T represents the right
Cauchy-Green deformation as usual. In particular, since the motion ξ , as well as the material
§ In standard crystal plasticity models, the number m of active glide system is determined among other
things by the glide-system “flow rule,” loading conditions, and crystal orientation. As such, it is constitutive
in nature, and in general variable.
214
B. Svendsen
velocity ξ˙, are not Euclidean frame-indifferent, is independent of these to satisfy material
frame-indifference. As such, (28) represents the basic reduced constitutive form of the constitutive class of interest for the continuum thermodynamic formulation of crystal plasticity to follow.
Because it plays no direct role in the formulation, the dependence of the constitutive relations on
p ∈ B is suppressed in the notation until needed.
The derivation of balance and field relations relative to the given reference configuration of
B is based in this work on the local forms for total energy and entropy balance, i.e.,
ė
=
div
η̇
=
π − div φ + σ ,
+s ,
(29)
respectively. Here, e represents the total energy density, its flux density, and s its supply rate
density. Likewise, π , φ , and σ represent the production rate, flux, and supply rate, densities,
respectively, of entropy, with density η. In particular, the mechanical balance relations follow
from (29)1 via its invariance with respect to Euclidean observer. And as usual, the thermodynamic analysis is based on (29)2 ; in addition, it yields a field relation for the temperature, as will
be seen in what follows.
This completes the synopsis of the basic relations required for the sequel. Next, we turn
to the formulation of field relations and the thermodynamic analysis for the constitutive class
determined by the form (28).
4. Generalized internal variable model for glide-system slips
The modeling of the γ as generalized internal variables (GIVs) is based in particular on the
standard continuum forms
e
(30)
s
=
ε
+
1 % ξ˙ · ξ˙ ,
2
=
−
+
T ξ˙ ,
=
r
+
· ξ˙ ,
for total energy density e, total energy flux density , and total energy supply rate density s ,
respectively, hold. Here, % represents the referential mass density, the first Piola-Kirchhoff
stress tensor, and the momentum supply rate density. Further, ε represents the internal energy
density, and the heat flux density. As in the standard continuum case, , ε, , η and φ
represent dependent constitutive quantities in general. Substituting the forms (30) for the energy
fields into the local form (29)1 for total energy balance yields the result
(31)
ε̇ + div − r =
· ∇ξ˙ − · ξ˙ + 21 c ξ˙ · ξ˙
for this balance. Appearing here are the field
(32)
c : = %̇
associated with mass balance, and that
(33)
: = ˙ − div − associated with momentum balance, where
: = % ξ˙
215
Non-local continuum thermodynamic
represents the usual continuum momentum density. As discussed by, e.g., Šilhavý (1997, Ch.
6), in the context of the usual transformation relations for the fields appearing in (31) under
change of Euclidean observer, one can show that necessary conditions for the Euclidean frameindifference of (29)1 in the form (31) are the mass
c=0
(34)
=⇒
%̇ = 0
via (32), momentum
=0
(35)
˙ = div + =⇒
via (33), and moment of momentum
T
(36)
=
balances, respectively, the latter with respect to the second Piola-Kirchhoff stress =
As such, beyond a constant (i.e., in time) mass density, we obtain the standard forms
˙
=
ε̇
=
(37)
0
1
2
· ˙
+
div +
,
−
div +
r ,
−1
.
for local balance of continuum momentum and internal energy, respectively, in the current context via (31), (35) and (36).
We turn next to thermodynamic considerations. As shown in effect by Maugin (1990), one
approach to the formulation of the entropy principle for material behaviour depending on internal
variables and their gradients can be based upon a weaker form of the dissipation (rate) inequality
than the usual Clausius-Duhem relation. This form follows from the local entropy (29) and
internal energy (37)2 balances via the Clausius-Duhem form
σ = r /θ
(38)
for the entropy supply rate σ density in terms of the internal energy supply rate density r and
temperature θ. Indeed, this leads to the expression
δ = 21
(39)
· ˙ − ψ̇ − η θ˙ + div (θφ − ) − φ · ∇θ
for the dissipation rate density
(40)
δ : = θπ
via (37)2 , where
ψ := ε − θ η
(41)
represents the referential free energy density. Substituting next the form (28) for ψ into (39)
yields that
(42)
δ
=
+
− ψ, } · ˙ − {η + ψ, θ } θ˙ − ψ, ∇ · ∇ θ˙ − ψ, γ˙ · γ¨ − ψ, ∇ γ˙ · ∇¨
γ
div (θφ − − 8VT γ˙ ) + ($V + div 8V ) · γ˙ − φ · ∇θ
{ 12
for δ via (27). Here,
(43)
$V : = −
T
ψ, α
,
216
B. Svendsen
$V : = ($V1 , . . . , $Vm ), and
8V : = T ψ, α
(44)
,
with 8V : = (ϕV1 , . . . , ϕVm ). Now, on the basis of (27) and (28), δ in (42) is linear in the fields
˙ , θ˙ , ∇ θ˙ , γ¨ and ∇¨γ . Consequently, the Coleman-Noll approach to the exploitation of the
entropy inequality implies that δ ≥ 0 is insured for all thermodynamically-admissible processes
iff the corresponding coefficients of these fields in (42) vanish, yielding the restrictions
(45)
=
2 ψ, ,
η
=
−ψ, θ ,
0
=
ψ, ∇ θ ,
0
=
ψ, γ̇ ,
0
=
ψ, ∇ γ̇ ,
= 1, . . . , m ,
= 1, . . . , m ,
on the form of the referential free energy density ψ , as well as the reduced expression
(46)
δ = div (θφ − − 8VT γ˙ ) + ($V + div 8V ) · γ˙ − θφ · ∇ln θ
for δ as given by (42), representing its so-called residual form for the current constitutive class.
In this case, then, the reduced form
ψ = ψ (θ, , α)
(47)
of ψ follows from (28) and (45).
On the basis of the residual form (46) for δ, assume next that, as dependent constitutive
quantities, $V + div 8V and φ are defined on convex subsets of the non-equilibrium part of the
state space, representing the set of all admissible ∇ θ, γ˙ and ∇˙
γ . If $V + div 8V and φ , again
as dependent constitutive quantities, are in addition continuously differentiable in ∇θ, γ˙ and ∇˙
γ
on the subset in question, one may generalize the results of Edelen (1973, 1985) to show¶ that
the requirement δ ≥ 0 on δ given by (46) yields the constitutive results
(48)
$V + div 8V
=
dV, γ˙ − div dV, ∇ γ˙ + ζV γ˙ ,
−θφ
=
dV, ∇ ln + ζV ∇ ln ,
for $V + div 8V and φ , respectively, in terms of the dissipation potential
dV = dV (θ, , α, ∇θ, γ˙ , ∇˙
γ)
(49)
and constitutive quantities
ζV γ˙
=
ζV γ˙ (θ, , α, ∇θ, γ˙ , ∇˙
γ),
ζV ∇ ln
=
ζV ∇ ln (θ, , α, ∇θ, γ˙ , ∇˙
γ),
which satisfy
(50)
ζV γ˙ · γ˙ + ζV ∇ ln · ∇ln θ = 0 ,
¶ In fact, this can be shown for the weaker case of simply-connected, rather than convex, subsets of the
dynamic part of state space via homotopy (see, e.g., Abraham et al. 1988, proof of Lemma 6.4.14).
217
Non-local continuum thermodynamic
i.e., they do not contribute to δ. To simplify the rest of the formulation, it is useful to work
with the stronger constitutive assumption that dV exists, in which case ζV γ˙ and ζV ∇ ln vanish
identically. On the basis then of the constitutive form
(51)
φ=
b θ −1 + θ −1 (8V + dV, ∇ γ˙ )T γ˙
for the entropy flux density, δ is determined by the form of dV alone, i.e.,
(52)
δ = dV, γ˙ · γ˙ + dV, ∇ γ˙ · ∇˙
γ + dV, ∇ ln · ∇ln θ
.
Among other things, (52) implies that a convex dependence of dV on the non-equilibrium fields
is sufficient, but not necessary, to satisfy δ ≥ 0. Indeed, with dV (θ, , α, 0, 0, 0) = 0, dV is
convex in ∇ θ, γ˙ and ∇˙
γ if δ ≥ dV (i.e., with δ given by (52)) for given values of the other
variables. So, if dV is convex in ∇ θ and γ˙ , and dV ≥ 0, then δ ≥ 0 is satisfied. On the other
hand, even if dV ≥ 0, δ ≥ 0 does not necessarily require δ ≥ dV , i.e., dV convex.
Lastly, in the context of the entropy balance (29)2 , the constitutive assumption (38), together
with (40) and the results (45)1,2 , (46) and (48), lead to the expression
(53)
c θ˙ = 21 θ , θ · ˙ + ωV + div dV, ∇ ln + r
for the evolution of θ via (47) and (49). Here,
c : = − θψ, θθ
(54)
represents the heat capacity at constant γ , , and so on, 21 θ , θ · ˙ = θψ, θ · ˙ the rate
(density) of heating due to thermoelastic processes, and
(55)
ωV : = (dV, γ˙ + θ
T
ψ, θα ) · γ˙ + (dV, ∇ γ˙ + θ T ψ, θα ) · ∇˙
γ
that due to inelastic processes via (27). In addition, (48)1 implies the result
(56)
dV, γ˙ = div ( T ψ, α + dV, ∇ γ˙ ) −
T
ψ, α
for the evolution of γ via (43) and (44). Finally,
(57)
− = dV, ∇ ln + ( T ψ, α + dV, ∇ γ˙ )T γ˙
follows for the heat flux density from (51) and (48)2 . As such, the dependence of ψ on α, as
well as that of dV on ∇˙
γ , lead in general to additional contributions to in the context of the
modeling of the γ as GIVs.
This completes the formulation of balance relations and the thermodynamic analysis for the
modeling of the γ as GIVs. Next, we carry out such a formulation for the case that the γ are
modeled as internal DOFs.
5. Internal degrees-of-freedom model for glide-system slips
Alternative to the model for the glide-system slips as GIVs in the sense of the last section is
that in which they are interpreted as so-called internal degrees-of-freedom (DOFs). In this case,
the degrees-of-freedomk of the material consist of (i), the usual “external” continuum DOFs
k This entails a generalization of the classical concept of “degree-of-freedom” to materials with structure.
218
B. Svendsen
represented by the motion ξ , and (ii), the “internal” DOFs γ . Or to use the terminology of
Capriz (1989), the γ are modeled here as scalar-valued continuum microstructural fields. Once
established as DOFs, the modeling of the γ proceeds by formal analogy with that of ξ , the only
difference being that, in contrast to external DOFs represented by ξ , each internal DOF γ is (i.e.,
by assumption) Euclidean frame-indifferent. Otherwise, the analogy is complete. In particular,
each γ is assumed to contribute to the total energy, the total energy flux and total energy supply,
of the material in a fashion formally analogous to ξ , i.e.,
e
(58)
s
=
ε
+
1 ˙
˙
2 ξ · %ξ
+
1 γ˙ · %I γ˙ ,
2
=
−
+
T ξ˙
+
8FT γ˙ ,
=
r
+
· ξ˙
+
ς · γ˙ ,
for total energy density e, total energy flux density
Here,

ι11 · · ·

 ..
..
I :=  .
.

ιm 1 · · ·
, and total energy supply rate density s .
ι1m
..
.
ιm m





is the (symmetric, positive-definite) matrix of microinertia coefficients, 8F : = (ϕF1 , . . . , ϕFm )
the array of flux densities, and ς : = (ς1 , . . . , ςm ) the array of external supply rate densities,
associated with γ . For simplicity, we assume that I is constant in this work. Next, substitution
of (58) into the general local form (29)1 of total energy balance yields
(59)
ε̇ + div − r =
· ∇ξ˙ + 8F · ∇˙γ − · ξ˙ − $F · γ˙ + 21 c (ξ˙ · ξ˙ + γ˙ · I γ˙ )
via (32) and (33). Here,
$F : = µ̇ − div 8F − ς
(60)
is associated with the evolution of γ ,
µ : = % I γ˙
(61)
being the corresponding momentum density. Consider now the usual transformation relations
for the field appearing in (58) and (59) under change of Euclidean observer, and in particular the
assumed Euclidean frame-indifference of the elements of γ , I, and 8F . As discussed in the last
section, using these, one can show that necessary conditions for the Euclidean frame-indifference
of (29)1 in the form (59) are the mass (34), momentum (35), and moment of momentum (36)
balances, respectively. As such, beyond a constant (i.e., in time) mass density, we obtain the set
(62)
˙
=
µ̇
=
ε̇
=
1
2
0
+
div +
,
$F
+
div 8F
+
ς ,
−
div +
r ,
· ˙ + 8F · ∇˙
γ − $F · γ˙
of field relations via (35), (36), (59) and (60).
Since we are modeling the γ as (internal) DOFs in the current section, the relevant thermodynamic analysis is based on the usual Clausius-Duhem constitutive forms
φ
=
/θ ,
σ
=
r /θ ,
(63)
219
Non-local continuum thermodynamic
for the entropy flux φ and supply rate σ densities, respectively. Substituting these into the
entropy balance (29)2 , we obtain the result
δ = 12
(64)
· ˙ + 8F · ∇˙
γ − $F · γ˙ − ψ˙ − η θ˙ − θ −1 · ∇θ
for the dissipation rate density δ : = θπ via (62)3 via (41). In turn, substitution of the constitutive
form (28) for the free energy ψ into (64), and use of that (27) for α, yields
δ
(65)
− ψ, } · ˙ − {η + ψ, θ } θ˙ − ψ,∇ θ · ∇θ˙ − θ −1 · ∇θ
=
{ 21
+
8FN · ∇˙
γ − $FN · γ˙ − ψ,γ˙ · γ¨ − ψ, ∇ γ˙ · ∇¨
γ ,
with
(66)
8FN
:=
8F − T ψ, α ,
$FN
:=
$F +
T
ψ, α ,
the non-equilibrium parts of 8F and $F , respectively. On the basis of (28), δ is linear in the
independent fields ˙ , θ˙ , ∇θ˙ , γ¨ and ∇¨
γ . As such, in the context of the Coleman-Noll approach to
the exploitation of the entropy inequality, δ ≥ 0 is insured for all thermodynamically-admissible
processes iff the corresponding coefficients of these fields in (65) vanish, yielding
(67)
=
2 ψ, ,
η
=
−ψ, θ ,
0
=
ψ, ∇ θ ,
0
=
ψ, γ̇ ,
0
=
ψ, ∇ γ̇ ,
= 1, . . . , m ,
= 1, . . . , m .
As in the last section, these restrictions also result in the reduced form (47) for ψ . Consequently,
the constitutive fields , ε and η are determined in terms of ψ as given by (47). On the other
hand, the 8FN , $FN as well as still take the general form (28). These are restricted further in
the context of the residual form
δ = 8FN · ∇˙
γ − $FN · γ˙ − θ −1 · ∇θ
for δ in the current constitutive class from (67). Treating 8FN , $FN and constitutively in
a fashion analogous to $V + div 8V and φ from the last section in the context of (48), the
requirement δ ≥ 0 results in the constitutive forms
(68)
8FN
=
dF, ∇ γ˙ + ζF ∇ γ˙ ,
−$FN
=
dF, γ˙ + ζF γ˙ ,
−
=
dF, ∇ ln + ζF ∇ ln ,
for these in terms of a dissipation potential dF and corresponding constitutive quantities ζF ∇ γ˙ ,
ζF γ˙ and ζF ∇ ln , all of the general reduced material-frame-indifferent form (28). As in the last
section, the latter three are dissipationless, i.e.,
(69)
ζF γ˙ · γ˙ + ζF ∇ γ˙ · ∇˙
γ + ζF ∇ ln · ∇ln θ = 0
220
B. Svendsen
analogous to (50) in the GIV case. Consequently, δ reduces to
δ = dF, γ˙ · γ˙ + dF, ∇ γ˙ · ∇˙
γ + dF, ∇ ln · ∇ln θ
via (68) and (69), analogous to (52) in the GIV case. In what follows, we again, as in the last
section, work for simplicity with the stronger constitutive assumption that dF exists, in which
case ζF γ˙ , ζF ∇ γ˙ and ζF ∇ ln vanish identically.
On the basis of the above assumptions and results, then, the field relation
(70)
c θ˙ = 21 θ , θ · ˙ + ωF + div dF, ∇ ln + r
for temperature evolution analogous to (53) is obtained in the current context via (54), with
(71)
ωF : = (dF, γ˙ + θ
T
ψ, θα ) · γ˙ + (dF, ∇ γ˙ + θ
T ψ, θα ) · ∇˙γ
the rate of heating due to inelastic processes analogous to ωV from (55). Finally, (68)1,2 lead to
the form
(72)
%I γ¨ + dF, γ˙ = div ( T ψ, α + dF, ∇ γ˙ ) −
T
ψ, α + ς
for the evolution of γ via (61), (62)2 and (66).
With the general thermodynamic framework established in the last two sections now in
hand, the next step is the formulation of specific models for GND development and their incoporation into this framework, our next task.
6. Effective models for GNDs
The first model for GNDs to be considered in this section is formulated at the glide-system level.
As it turns out, this model represents a three-dimensional generalization of the model of Ashby
(1970), who showed that the development of GNDs in a given glide system is directly related
to the inhomogeneity of inelastic deformation in this system. In particular, in the current finitedeformation context, this generalization is based on the incompatibility of with respect to the
reference placement. To this end, consider the vector measure∗∗
I
I
(C)
:
=
(73)
γ ( · C ) =
G
C
C
C
of the length of glide-system GNDs around an arbitrary closed curve or circuit C in the reference
configuration, the second form following from (20). Here, C represents the unit tangent to C
oriented clockwise. Alternatively, G (C) is given by††
Z
I
(74)
(curl ) S
(C)
:
=
=
G
C
C
S
with respect to the material surface S bounded by C via Stokes theorem. Here,
(75)
curl = ( ⊗ )( × ∇γ ) = ⊗ (∇γ × )
∗∗ Volume dv, surface da and line d` elements are suppressed in the corresponding integrals appearing
in what follows for notational simplicity. Unless otherwise stated, all such integrals to follow are with
respect to line, surfaces and/or parts of the arbitrary global reference placement of the material body under
consideration.
†† Note that curl appearing in (74) is consistent with the form (8) for the curl of a second-order Euclidean tensor field.
221
Non-local continuum thermodynamic
from (7), (20), and the constancy of ( , , ). On the basis of (74), G (C) can also be
interpreted as a vector measure of the total length of GNDs piercing the material surface S
enclosed by C. The quantity curl determines in particular the dislocation density tensor α (I )
worked with recently
on the incompatibility
of their slip
Pn
Pn by Shizawa and Zbib (1999) as based
tensor γ (I ) : =
=1 γ ⊗ . Indeed, we have α(I ) : = curl γ (I ) = =1 curl in the
current notation.
Now, from (73) and the constancy of , note that G (C) is parallel to the slip direction ,
i.e.,
G (C) = lG with
(76)
lG (C) : =
I
C
γ · C =
Z
S
(curl )T · S
the scalar length of GNDs piercing S via (74). With the help of a characteristic Burgers vector
magnitude b, this length can be written in the alternative form
Z
(77)
lG (C) = b
G · S
S
in terms of the vector field
(78)
G
G
determined by
: = b −1 (curl )T = b −1 ∇γ × .
From the dimensional point of view, G represents a (vector-valued) GND surface (number)
density. As such, the projection G · S of G onto S gives the (scalar) surface (number) density
of such GNDs piercing S. The projection of (78) onto the glide-system basis ( , , ) yields
(79)
·
G
=
−b −1 · ∇γ ,
·
G
=
b −1 · ∇γ ,
·
G
=
0,
for the case of constant b. In particular, the first two of these expressions are consistent with
two-dimensional results of Ashby (1970) for the GND density with respect to the slip direction
and that perpendicular to it in the glide plane generalized to three dimensions. Such threedimensional relations are also obtained in the recent crystallographic approach to GND modeling
of Arsenlis and Parks (1999). Likewise in agreement with the model of Ashby (1970) is the
fact that (79)3 implies that there is no GND development perpendicular to the glide plane (i.e.,
parallel to ) in this model. ¿From another point of view, if ∇γ were parallel to , there
would be no GND development at all in this model; indeed, as shown by (75), in this case, would be compatible.
The second class of GND models considered in this work is based on the vector measure
Z
I
(curl P ) S
(C)
:
=
(80)
=
G
P C
C
S
of the length of GNDs from all glide systems around C in the material as measured by the
incompatibility of the local inelastic deformation P . In particular, the phenomenological GND
model of Dai and Parks (1997), utilized by them to model grain-size effects in polycrystalline
metals, applied as well recently by Busso et al. (2000) to model size effects in nickel-based
superalloys, is of this type. In a different context, the incompatibility of P has also been used
222
B. Svendsen
recently by Ortiz and Repetto (1999), as well as by Ortiz et al. (2000), to model in an effective
fashion the contribution of the dislocation self- or core energy to the total free energy of ductile
single crystals. In what follows, we refer to the GND model based on the measure (80) as
the continuum (GND) model. To enable comparison of this continuum GND model with the
glide-system model discussed above, it is useful to express the former in terms of glide-system
quantities formally analogous to those appearing in the latter. To this end, note that the evolution
relation (24) for P induces the glide-system decomposition
Xn
G (C) =
lG (C) =1
of G (C) in terms of the set lG1 (C), . . ., lGn (C) of glide-system GND lengths with respect to C
formally analogous to those (76) in the context of the glide-system GND model. In contrast to
this latter case, however, each lG here is determined by an evolution relation, i.e.,
I
Z
(81)
l˙G (C) =
γ˙ PT · C =
curl ( ˙ P )T · S ,
C
with
γ ) + ⊗ (curl P )T γ˙ )( × ∇˙
via (7) and (20). Alternatively, we can express lG (C) as determined by (81) in the form (77)
involving the vector-valued GND surface density G , with now
(82)
curl ( ˙
S
P)
= ( ⊗
˙G = b −1 curl ( ˙
P)
T
T
P
γ ×
= b −1 ∇˙
T
P
+ b −1 (curl
P)
T
γ˙ in the context of (80). As implied by the notation, ˙G from (82) in the current context is formally
analogous to the time-derivative of (78) in the glide-system GND model. Now, from the results
(26) and (82), we have
Xm
˙
curl P = b
⊗ ˙
=1
and so the expression‡‡
curl
P
=b
Xn
=1 ⊗ for the incompatibility of P in terms of the set ( 1 , . . . , n ) of vector-valued GND densities.
Substituting this result into (82) then yields
X
−1 γ × T ˙G =
P
( · ) G γ˙ + b ∇˙
6=
with · = 0 and
P
6= : =
Pm
=1, 6 = . Relative to ( , , ), note that
· ˙G =
b −1
T
P
× · ∇˙
γ
+
· ˙G =
b −1
T
P
× · ∇˙
γ
+
=
b −1
T
P
γ
× · ∇˙
+
· ˙G X
( · ) ·
G
γ˙ ,
( · ) ·
G
γ˙ ,
( · ) ·
G
γ˙ ,
X 6=
X
6= 6= via (8) and (21), analogous to (79). In contrast to the glide-system GND model, then, this
approach does lead to a development of (edge) GNDs perpendicular to the glide plane (i.e.,
parallel to ).
‡‡ Assuming the integration constant to be zero for simplicity, i.e., that there is no initial inelastic incompatibility.
223
Non-local continuum thermodynamic
To summarize then, we have the expressions

−1 ˙

 b ∇γ × (83) ˙G =
X

 b −1 ∇γ˙ × T +
P
( · )
6=
glide-system model
G
γ˙ continuum model
for the evolution of the vector-valued measure G of GND density from the glide-system and
continuum models discussed above. With these in hand, we are now ready to extend existing
models for crystal plasticity to account for the effects of GNDs on their material behaviour, and in
particular their effect on the hardening behaviour of the material. In the current thermodynamic
context, such extensions are realized via the constitutive dependence of the free energy on GND
density, and more generally on the dislocation state in the material, our next task.
7. Free energy and GNDs
With GND models such as those from the last section in hand, the question arises as to how these
can be incoporated into the thermodynamic formulation for crystal plasticity developed in the
previous sections. Since this formulation is determined predominantly by the free energy density
ψ and dissipation potential d , this question becomes one of (i), which quantities characterize
effectively (i.e., phenomenologically) the GND, and more generally dislocation, state of each
material point p ∈ B , and (ii), how do ψ and d depend on these? The purpose of this section is
to explore these issues for the case of the referential free energy density ψ . In particular, this
involves the choice for α.
Among the possible measures of the inelastic/dislocation state of each material point, we
have the arrays ρS = (ρS1 , . . . , ρSn ) and G = ( G1 , . . . , Gn ) of glide-system effective SSD
and GND densities, respectively. Choosing then α = ( P , ρS , G ), ψ takes the form
ψ (θ, , α) = ψD (θ, ,
P , ρS ,
G)
for ψ from (47), with
ρ̇S (84)
˙G Xm
=
=1
Xm
=
KS γ˙ ,
=1
G
γ ,
γ˙ + G ∇˙
from (27). This choice induces the decompositions
T
(85)
T
ψ, α
ψ, α
=
=
( ˙P, γ˙ )T ψD, P
0
+
+
T
S ψD, ρS
0
+
+
T
G ψD, G
,
T
G ψD, G
,
from (24) of the constitutive quantities T ψ, α and T ψ, α determining the form (56) or (72) of
the field relation for γ . In particular, the models (83) for ˙G yield

glide-system model
 0
X
=
(86)
G
 δ
continuum model
( · ) G 6=
and
(87)
G = b −1 δ (
×
×
T
P
glide-system model
continuum model
224
for
B. Svendsen
G
(88)
and G , respectively. And from (87), we have
( ψ, α ) = ( T
T
G
ψD, ) = b −1
G
(
× ψD,
T
P
glide-system model
× ψD,
continuum model
for the flux contribution appearing in the evolution relation (56) or (72) for γ . From (84), (85)1
and (86), we have
( T ψ, α ) = −τ + x + sgn(γ˙ ) r
where
(89)
τ = −(( ˙P, γ˙ )T ψ, ) = −( ⊗ )
P
represents the glide-system Schmid stress via (24),

 0
X
(90)
x : =

·
( · ) ψD,
6=
G
P
· ψ, P
glide-system model
continuum model
G
(a contribution to) the glide-system back stress, and
Xm
IS ψD, ρ
(91)
r : =
=1
S
the glide-system yield stress, with KS = IS sgn(γ˙ ). Note that sgn(γ˙ ) is a constitutive
quantity in existing crystal plasticity models. For example, in the case of the (non-thermodynamic) glide-system flow rule
τ n
sgn(τ )
(92)
γ˙ = γ˙ 0 τC of Teodosiu and Sideroff (1976) (similar to the form used by Asaro and Needleman, 1985; see
also Teodosiu, 1997), we have sgn(γ˙ ) =
b sgn(τ ). Here, τC represents the critical Schmid
stress for slip. In particular, such a constitutive assumption insures that the contribution τ γ˙ =
|τ ||γ˙ | = |τ |ν̇ to the dissipation rate density remains greater than or equal to zero for all
∈ {1, . . . , m }. Such a constitutive assumption is made for other types of glide-system flow
rules, e.g., the activation form
1G (|τ |, τC )
γ˙ = γ˙ 0 exp −
sgn(τ )
kB θ
used by Anand et al. (1997) to model the inelastic behaviour of tantalum over a much wider
range of strain rates and temperatures than possible with (92). Here, 1G (|τ |, τC ) represents
the activation Gibbs free energy for thermally-induced dislocation motion.
Consider next the dependence of ρ˙S on γ˙ , i.e., KS . As it turns out, a number of existing
approaches model this dependence. For example, in the approach of Estrin (1996, 1998) to
dislocation-density-based constitutive modeling (see also ∗ Estrin et al., 1998; Sluys and Estrin,
2000), this dependence follows from the constitutive relation
o
nX n
q
(93)
ρ̇S =
jS ρS − kS ρS sgn(γ˙ ) γ˙ =1
∗ Because their model for SSD flux includes a Fickian-diffusion-like contribution due to dislocation
cross-slip proportional to · ∇ρS
, the approach of Sluys and Estrin (2000) does not fit into the current
framework as it stands. The necessary extension involves treating the SSD densities ρS as, e.g., (independent)
GIVs, analogous to the γ .
225
Non-local continuum thermodynamic
for the evolution of ρS in terms of magnitude |γ˙ | = sgn(γ˙ ) γ˙ of γ˙ for = 1, . . . , m . In
(93), jS11 , jS12 , . . . represent the elements of the matrix of athermal dislocation storage coefficients, which in general are functions of ρS , and k the glide-system coefficient of thermallyactivated recovery. In this case, then,
nX n
o
p
IS = δ jS ρS − kS ρS
=1
holds, and so
(
T
S ψD, ρS )
= KS ψD, ρ
= sgn(γ˙ ) IS ψD, ρ
S
S
from (91). Other such models for KS can be obtained analogously from existing ones for ρ̇S in the literature, e.g., from the dislocation-density-based approach of Teodosiu (1997).
Models such as those (83)2 for G , or that (93) in the case of ρS , account in particular for
dislocation-dislocation interactions. At least in these cases, then, such interactions are taken into
account in the evolution relations for the dislocation measures, and so need not (necessarily) be
accounted for in the form of ψ . From this point of view, ψD could take for example the simple
“power-law” form
Xn
Xn
2g
−1
2s
ψD = 21 E · E E + s −1 cS µ
=1 S + g cG µ =1 G (94)
=:
ψDE
+
ψDS
+
ψDG
in the case of ductile single crystals, perhaps the simplest possible. Here,
referential elasticity tensor, and
1
E : = 2 ( E − )
E represents the
the elastic Green strain determined by the corresponding right Cauchy-Green tensor
E
(95)
:=
−T
P
−1
P
.
Further, cS and cG are (scaling) constants, s and g exponents, µ the average shear modulus, and
p
: = `S ρS ,
S q
G : = `G | G | ,
non-dimensional deformation-like internal variables associated with SSDs and GNDs, respectively, involving the characteristic lengths `S and `G , respectively. In particular, the GND contribution ψDG to ψD appearing in (94) is motivated by and represents a power-law generalization of
the model of Kuhlmann-Wilsdorf (1989) for dislocation self-energy (see also Ortiz and Repetto,
1999) as based on the notion of dislocation line-length. From (94), we have in particular the
simple expression
τ = ⊗ · 2 E ψDE, E
for the Schmid stress τ from (89) in terms of the Mandel stress −ψD, P
addition,
s−1
2s
ψD, ρ
= cS µ `S ρS ,
ψD,
=
2g
cG µ `G |
then hold. From these, we obtain in turn

 0
X
(96) x =
 cG µ `G2g | G |g−2
( · )
6= G
|g−2
G
T
P
= 2 E ψDE, . In
,
glide-system model
G
·
G
continuum model
E
226
B. Svendsen
from (90) for x ,
(97)
(
T
G
2g
ψD, ) = cG µ b −1 `G |
G
G
|g−2
(
×
T
P
G
×
glide-system model
G
continuum model
from (88), as well as the result
(98)
r = cS µ `S2s
Xm
=1
IS ρSs−1
from (91) for r . On the basis of models like that (93) of Estrin (1998) for ρ˙S , this last form for r
is consistent with and represents a generalization of distributed dislocation strength models (e.g.,
Kocks, 1976, 1987) to account for the effectsqof GNDs on glide-system (isotropic) hardening.
Indeed, for s = 1, r becomes proportional to ρS in the context of (93).
The simplest case of the formulation as based on (94) arises in the context of the glidesystem model for GNDs when we set s = 1 and g = 2. Then
( −1
b × (∇γ × )
glide-system model
T
4
−1
(99)
( G ψD, ) = cG µ `G b
T
G
continuum model
P × G follows from (87) for the flux contribution via (78) and (83). Note that (99)1 follows from the
fact that (83)1 is integrable. Then, the corresponding reduction of r from (98), (56) and (99)1
implies in particular the evolution-field relation
Xm
(100)
dV, γ̇ = cG µ `G4 b −2 div (∇γ ) + τ − cS µ `S2
KS =1
for γ modeled as a GIV via (56), again in the context of the glide-system GND model, assuming
cG , µ and `G constant. The perhaps simplest possible non-trivial form of (100) for the evolution
of γ in the current context follows in particular from the corresponding simplest (i.e., quasilinear) form∗ dV, γ̇ = β γ˙ for dV, γ̇ in terms of the glide-system damping modulus β ≥ 0
with units of J s m−3 or Pa s (i.e., viscosity-like). In addition,
div (∇γ ) : = ( − ⊗ ) · ∇(∇γ ) = ( ⊗ + ⊗ ) · ∇(∇γ )
represents the projection of the divergence operator onto the glide plane spanned by ( , ).
Given suitable forms for the constitutive quantities, then, the field relation (100) can in principal
be solved (i.e., together with the momentum balance in the isothermal case) for γ . On the other
hand, since P does not depend explicitly on γ , and, in contrast to the glide-system model, G does not depend explicitly on γ and ∇γ in the continuum GND model, no “simple” expression
like (100) for the evolution of γ is obtainable in this case. Indeed, in all other cases, one must
proceed more generally to solve initial-boundary-value problems for ξ , the γ , and θ. We return
to this issue in the next section.
A second class of free energy models can be based on the choice α = ( P , ν , curl P ), i.e.,
(101)
with
ψ (θ, , α) = ψC (θ, ,
P , ν , curl
P)
,
ν̇ = |γ˙ |
∗ The coupling with ∇ in d and the dependence of d on ∇γ˙ , is neglected here.
V
V
227
Non-local continuum thermodynamic
the glide-system accumulated slip rate. In this case, we have
ψ, α ) =
( T ψ, α ) =
(
(102)
T
−τ + x + sgn(γ˙ ) r ,
T
P
× (ψC, curl )T ,
P
via (26) and (89), where now
x : = ⊗ · ψC, curl (curl
(103)
P
and
P)
T
r : = ψC, ν
Consider for example the particular form
(104)
=
ψC
=:
1
2
E
· E
E
ψCE
+
ψCS (ν )
+
+
ψCS
+
g
g −1 cG µ `G |curl
P|
g
ψCG
for ψC analogous to (94) for ψD . In this context, the choice
"
!#
(τ − τ0 )2
h0
ψCS = S
ln cosh
ν
h0
τS − τ0
yields the simple model for isotropic or Taylor hardening (i.e., due to SSDs) proposed by Hutchinson (1976), with
Xn
ν :=
ν
=1
the total accumulated inelastic slip in all glide systems. Here, τS represents a characteristic
saturation strength, τ0 a characteristic initial critical resolved shear stress, and h0 a characteristic
initial hardening modulus, for all glide systems. Another possibility for ψCS is the form
Xn
−1
s
ψCS =
r0 ν + s h0 ν
=1
consistent with the model of Ortiz and Repetto (1999) for latent hardening in single crystals,
with now
r
Xn
ν :=
ıS ν ν
, =1
the effective total accumulated slip in all glide systems in terms of the interaction coefficients
ıS , , = 1, . . . , n . In particular, this model is based on the assumptions that (i), hardening is
parabolic in single slip (i.e., for s = 2/3), and (ii), the hardening matrix ψCS, ν ν is dominated
by its off-diagonal components. Beyond such models for glide-system (isotropic) hardening,
(104) yields the expression
g
x = cG µ `G |curl
P|
g−2
⊗ · (curl
P )(curl
P)
T
for glide-system back-stress from (103), as well as that
( T ψ, α ) = cG µ `G |curl
g
P|
for ( GT ψC, curl ) from (102)2 . Analogous to
g−2
T
P
× (curl
P)
T
in the case of the continuum GND
model, because P and curl P do not depend explicitly on γ and ∇γ , no field relation for γ of
P
P
and
G
the type (100) follows from (104), and we are again forced to proceed numerically.
228
B. Svendsen
8. The case of small deformation
Clearly, the formulation up to this point is valid for large deformation. For completeness, consider in this section the simplifications arising in the formulation under the assumption of small
deformation. In particular, such a simplification is relevant to comparisons of the current approach with other modeling approaches such as the dislocation computer simulation of Van der
Giessen and Needleman (1995). This has been carried out recently (Svendsen & Reese, 2002)
in the context of the (isothermal) simple shear of a crystalline strip containing one or two glide
planes. This model problem has been used in the recent work of Shu et al.(2001) in order to
compare the predictions of the discrete dislocation computer simulation with those of the nonlocal strain-gradient approach of Fleck and Hutchinson (1997) and applied to crystal plasticity
(e.g., Shu and Fleck, 1999). As dicusssed by them, it represents a model problem for the type of
plastic constraint found at grain boundaries of a polycrystal, or the surface of a thin film, or at
interfaces in a composite.
In the crystal plasticity context, the small-deformation formulation begins with the corresponding form
Xn
(105)
P =
( ⊗ ) γ
=1
for the local inelastic displacement “gradient” P assuming no initial inelastic deformation
in the material. Note that this measure is in effect equivalent to the slip tensor γ (I ) : =
Pn
=1 γ ⊗ of Shizawa and Zbib (1999). In addition, note that P can be considered
as a function of γ in this case. In turn, (105) yields the expression
Xn
(106)
curl P =
⊗ (∇γ × )
=1
for the incompatibility of P . This is equivalent to the dislocation density tensor α (I ) : =
curl γ (I ) of Shizawa and Zbib (1999). Note that either curl P or this latter measure may be
considered a function of ∇γ . In this context, then, rather than for example with the choice
( P , ν , curl P ), we could work alternatively with that α = (γ , ν , ∇γ ) as a measure for the
inelastic/dislocation state in the material at any material point p ∈ B . In fact, it would appear
to be the simplest possible choice. Indeed, any such choice based alternatively on the smalldeformation form

 b −1 ∇γ˙ × glide-system model
˙G =
 b −1 ∇γ˙ × +
X
6= ( · )
G
γ˙ continuum model
of (83) for the development of vector-valued glide-system GND density G would appear, at
least in the context of the continuum model, to be more complicated since ˙G is not exactly
integrable, i.e., even in the small-strain case. On this basis, the general constitutive form (28)
reduces to
= (θ, , γ , ν , ∇γ , ∇θ, γ˙ , ∇˙γ , p)
for all dependent constitutive quantities (e.g., stress) in the small-deformation context, again with
α = (γ , ν , ∇γ ). Here,
: = sym(∇ )
represents the symmetric part of the displacement gradient. By analogy, the results of the thermodynamic formulations in §§4–5 for γ modeled as GIVs or as internal DOFs can used to obtain
those for the case of small strain. Further, the reduced form (47) of ψ becomes
ψ = ψ (θ,
, γ , ν , ∇γ ).
229
Non-local continuum thermodynamic
Consider for example the class
(107)
ψ (θ,
, γ , ν , ∇γ ) = ψC (θ,
,
P (γ
), ν , curl
P (∇γ
))
of forms for ψ analogous to (101), with
(108)
P
: = sym(
P)
the inelastic strain. From (107) follow
ψ, α ) =
ψ, γ + sgn(γ˙ ) ψC, ν
=
−τ + sgn(γ˙ ) ψC, ν ,
( ψ, α ) =
ψ, ∇ γ
=
× (ψC, curl
(
T
T
P
)T ,
by analogy with (102) via (108) and (106), with now
τ : = − · ψC,
P
for the Schmid stress. In the case of small deformation, then, the contribution x from inhomogeneity to the glide-system back stress vanishes identically. On the basis of these results, the
form
(109)
dV, γ̇ = div [ × (ψC, curl
P
)T + dV, ∇ γ̇ ] + τ − sgn(γ˙ ) ψC, ν
of the evolution relation for the γ from (56) in the context of their modeling as GIVs, holds.
Further insight into (109) can be gained by introducing concrete forms for ψC and dV . For
example, consider that
(110)
ψC = 21
E
· E
E
+ ψCS (ν ) + g −1 µ `g |curl
P|
g
for ψC analogous to (104), with E : = − P now the (small) elastic strain, and
sym( P ) the (small) inelastic strain. Further, the power-law form
(111)
P
:=


Xm  |γ˙ | (n +1)/n
n



dV =
ς ν̇0

=1  ν̇ 
n +1
0
for the dissipation potential dV is perhaps the simplest one for dV of practical relevance. Here, ς
represents a characteristic energy scale for activation of dislocation glide motion with units of J
m−3 or Pa, and ν̇0 a characteristic value of |γ˙ |. Substituting (110) and (111) into (109) results
in the evolution/field relation
Xm
ς ν̇0−1 γ˙ = µ `2
· ∇(∇γ ) + · ( E E ) − sgn(γ˙ ) ψCS, ν
=1
for the glide system slip γ via the g = 2 and n = 1, with : = ( · ) [( · ) − ⊗
]. In particular, note that ·∇(∇γ ) = [ − ⊗ ]·∇(∇γ ) = [ ⊗ + ⊗ ]·∇(∇γ )
represents the divergence of ∇γ projected onto the th glide plane. It is worth emphasizing that
the form of this projection results from the dependence of ψ on curl P . For comparison, note
that : = ( · ) ( · ) , and so · ∇(∇γ ) = ( · ) ( · ) div(∇γ ), would
hold if ψ depended on the inhomogeneity ∇ P instead of on the incompatibility curl P of P .
In the crystal plasticity and current context, at least, the distinction is significant in the sense that
no additional hardening results in the current context when ψ depends directly on ∇ P .
230
B. Svendsen
9. Discussion
Consider the results of the two approaches to the modeling of the glide-system slips γ from
§§4-5. Formally speaking, these differ in (i), the respective forms (56) and (72) for the evolution
of the γ , (ii), those (55) and (71) for the rate of heating ω due to inelastic processes, and (iii),
those (57) and (68)3 for the heat flux density . In particular, in view of the corresponding forms
(53) and (70) for temperature evolution, this latter difference is of no consequence for the field
relations. Indeed, except for the contribution ζF∇ γ˙ to 8N in the internal DOF model for the γ ,
the total energy flux density has the same form in both cases, i.e.,
=
− + T ξ˙ = dV, ∇ ln + ζV ∇ ln + ( T ψ, α + dV, ∇ γ˙ )T γ˙ + T ξ˙ ,
=
− + T ξ˙ + 8FT γ˙ = dF, ∇ ln + ζF ∇ ln + ( T ψ, α + dF, ∇ γ˙ + ζF∇ γ˙ )T γ˙ + T ξ˙ ,
from (30)2 , (57), (66)1 and (68)1 . This fact is related to the observation of Gurtin (1971, footnote
1) in the context of classical mixture theory concerning the interpretation of “entropy flux” and
“heat flux” in phenomenology and the relation between these two. There, the issue was one of
whether diffusion flux is to be interpreted as a flux of energy (e.g., Eckhart, 1940; Gurtin, 1971)
or a flux of entropy (e.g., Meixner and Reik, 1959; DeGroot and Mazur, 1962; Müller, 1968).
In the current context, the flux (density) of interest is that ( T ψ, α + d, ∇ γ˙ )T γ˙ . In the GIV
approach, the constitutive form (51) shows that this flux (i.e., divided by θ) is being interpreted
as an entropy flux. On the other hand, (58)2 and (63)2 imply that it is being interpreted as an
energy flux in the internal DOF approach. In this point, then, both approaches are consistent
with each other.
As it turns out, the field relation (56) for the γ derived on the basis of the modeling of these
as generalized internal variables, represents a generalized form of the Cahn-Allen field relation
(e.g., Cahn, 1960; Cahn and Allen, 1977) for non-conservative phase fields, itself in turn a generalization of the Ginzburg-Landau model for phase transitions. In particular, (56) would reduce to
the Cahn-Allen form (i), if dV were proportional to a quadratic form in γ˙ and independent of ∇˙
γ,
and (ii), if T ψ, α and T ψ, α were reduceable to ψ, ∇ γ and ψ, γ , respectively. In particular,
this latter case arises only for monotonic loading and small deformation. The Cahn-Allen relation has been studied quite extensively from the mathematical point of view (see, e.g., Brokate
and Sprekels, 1996). As such, one may profit from the corresponding literature on the solution
of specific initial-boundary value problems in applications of the approach leading to (56), or
more generally that leading to (72), which are currently in progress.
From a phenomenological point of view, the concrete form (94) for ψD , and in particular
that of ψDE , or that of ψCE in (104), is contingent upon the modeling of P as an elastic material isomorphism (e.g., Wang and Bloom, 1974; Bertram, 1993; Svendsen, 1998), i.e., inelastic
processes represented by P do not change the form of the elastic constitutive relation. Such an
assumption, quite appropriate and basically universal for single-crystal plasticity, may be violated in the case of strong texture development, induced anisotropy and/or anisotropic damage
in polycrystals. As discussed by Svendsen (1998), one consequence of the modeling of P as an
elastic isomorphism is the identification of
(112)
E
:=
−1
P
as the local (elastic) deformation in the material, and in particular that of the crystal lattice in
single-crystal plasticity. More generally, P can be modeled as a material uniformity (Maugin
and Epstein, 1998; Svendsen, 2001b) in the case of simple materials. In the current context,
231
Non-local continuum thermodynamic
(112) implies the connection
curl P
E
= −det( P−1 )
E (curl
T
P) P
via (9) and (8) between incompatibility of the local lattice deformation E with respect to the
intermediate (local) “configuration” and that of P with respect to the reference (local) “configuration” (i.e., placement) at each p ∈ B via (10), (11), and the compatibility of . Alternatively,
we have
curlP E = − E I ,
where∗
I : = det(
(113)
P)
−1 (curl
P)
T
P
= det(
E ) (curl
−1
E
) E−T
represents the geometric dislocation tensor recently introduced by Cermelli and Gurtin (2001).
As shown by them, I represents the incompatibility of P relative to the surface element
I daI : = det( P )
−T
P
S da S
in the intermediate configuration. Indeed, relative to this element, the equivalence
(curl
P)
S da S =
I I daI
holds. As such, curl P gives the same measure of GNDs with respect to surface elements in the
reference configuration as does I with respect to such elements in the intermediate configuration. Note that I , like curl P , has units of inverse length. The definition (113)1 implies the
form
˙ = curlP + P
P I +
I
(114)
for the evolution of
(115)
det(
P)
I P −
I ( ·
T
P )
I via (22). Alternatively, this can be expressed “objectively” as
−1
P
[det(
P)
˙−1
−T
I
P
P
T
P
]
= det(
: = det(
−1
P)
P
−T
I
P
−1
P
˙
R
T
P
I with respect to
relative to the “upper” Oldroyd-Truesdell derivative of
R
P)
=
−1
P
(curl
= curl P P
P,
where
P)
represents the referential form of I via (113). In the current crystal plasticity context, the
right-hand side of (115) reduces to
Xn
Xn
−T
−T
curl P P = b
γ × ]
⊗ P ˙G =
⊗ P [∇˙
=1
=1
via (6) and (23) in terms of the evolution of the vector-valued GND surface density G for the
glide-system GND model from (78). As such, (114) implies
Xm
Xm
−T
˙ =
γ × ]
I
[( ⊗ ) I + I ( ⊗ )] γ˙ + ⊗ P [∇˙
=1
=1
for the evolution of I in the case of crystal plasticity via (23) and the fact that · P = 0 in
this context. So, another class of specific forms for ψ from (47) can be based on the choice
α = ( P , ν , I , implying
ψ (θ, , α, p) = g(θ, E , ν , I , p)
∗ Recall that we have defined the curl of a second-order tensor field in (6) via (curl )T : = curl ( T ),
rather than in the form (curl ) : = curl ( T ) used by Cermelli and Gurtin (2001).
232
B. Svendsen
via (95). In turn, g itself is a member of the class defined by the choice α = ( P , ν , ∇ P ), as
can be concluded directly from (25) and the fact that curl P is a function of ∇ P via (18). As
shown by Cermelli and Gurtin (2001), constitutive functions for any p ∈ B depending on ∇ P
must reduce to a dependence on I for their form to be independent of change of compatible
local reference placement at p ∈ B , i.e., one induced by a change∗ of global reference placement.
This requirement is in turn based on the result of Davini (1986), and Davini and Parry (1989) that
such changes leave dislocation measures such as I unchanged, representing as such “elastic”
changes of local reference placement. As it turns out, one can show more generally (Svendsen,
2001c) that ψ reduces to g for all p ∈ B , i.e., for B as a whole, under the assumption that P
represents a particular kind of material uniformity.
References
[1] A BRAHAM R., M ARSDEN J.E. AND R ATIU T., Manifolds, tensor analysis and applications, Appl. Mat. Sc. 75, Springer, New York 1988.
[2] ACHARYA A. AND BASSANI J.L., Lattice incompatibility and a gradient theory of crystal
plasticity, J. Mech. Phys. Solids 48 (2000), 1565–1595.
[3] ACHARYA A. AND B EAUDOIN A.J., Grain-size effect in viscoplastic polycrystals at moderate strains, J. Mech. Phys. Solids 48 (2000), 2213–2230.
[4] A NAND L., BALASUBRAMANIAN S. AND KOTHARI M., Constitutive modeling of polycrystalline metals at large strains, application to deformation processes; in: “Large plastic
deformation of crystalline aggregates” (Ed. C. Teodosiu), CISM 376, Springer 1997, 109–
172.
[5] A RSENLIS A. AND PARKS D.M., Crystallographic aspects of geometrically-necessary
and statistically-stored dislocation density, Acta Mater. 47 (1999), 1597–1611.
[6] A SARO R.J., Micromechanics of crystals and polycrystals, Adv. Appl. Mech. 23 (1983),
1–115.
[7] A SARO R.J. AND N EEDLEMAN A., Texture development and strain hardening in ratedependent polycrystals, Acta Metall. 33 (1985), 923–953, 1985.
[8] A SHBY M.F., The deformation of plastically non-homogeneous materials, Phil. Mag. 21
(1970), 399–424.
[9] B ERTRAM A., Description of finite inelastic deformations; in: “MECAMAT ’92 Multiaxial
Plasticity” (Eds. A. Benallal, R. Billardon and D. Marquis), 1993, 821–835.
[10] B ROKATE M. AND S PREKELS J., Hysteresis and phase transitions, App. Mat. Sciences
121, Springer, New York 1996.
[11] B USSO E.F., M EISSONNIER F.T. AND O’D OWD N.P., Gradient-dependent deformation
of two-phase single crystals, J. Mech. Phys. Solids 48 (2000), 2333–2361.
[12] C AHN J.W., Theory of crystal growth and interface motion in crystalline materials, Acta
Metall. 8 (1960), 554–562.
[13] C AHN J.W. AND A LLEN S.M., A microscopic theory for domain wall motion and its
experimental verification in Fe-Al alloy domain growth kinetics, J. de Physique 38 (1977),
51–54.
∗ Since local changes of reference placement for any p ∈ B represent equivalence classes of corresponding changes of global reference placement there, any local change of reference placement for p ∈ B is induced by a corresponding change of global placement.
Non-local continuum thermodynamic
233
[14] C APRIZ G., Continua with microstructure, Tracts in Natural Philosophy 37, Springer, New
York 1989.
[15] C ERMELLI P. AND G URTIN M.E., On the characterization of geometrically necessary
dislocations in finite plasticity, J. Mech. Phys. Solids (2001).
[16] C UITI ÑO A.M. AND O RTIZ M., Computational modeling of single crystals, Modelling
Simul. Mater. Sci. Eng. 1 (1992), 225–263.
[17] DAI H. AND PARKS D.M., Geometrically-necessary dislocation density and scaledependent crystal plasticity; in: “Proceedings of Plasticity” (Ed. A.S. Khan), Neat Press
1997, 17–18.
[18] DAVINI C., A proposal for a continuum theory of defective crystals, Arch. Rat. Mech. Anal.
96 (1986), 295–317.
[19] DAVINI C. AND PARRY G.P., On defect-preserving deformations in crystals, Int. J. Plast.
5 (1989), 337–369.
[20] D E G ROOT S.R.
1962.
AND
M AZUR P., Non-equlibrium thermodynamics, Wiley-Interscience
[21] E CKHART C., The thermodynamics of irreversible processes; II. Fluid mixtures, Phys. Rev.
58 (1940), 269–275.
[22] E DELEN D.G.B., On the existence of symmetry relations and dissipation potential, Arch.
Rat. Mech. Anal. 51 (1973), 218–227.
[23] E DELEN D.G.B., Applied exterior calculus, John Wiley & Sons, New York 1985.
[24] E STRIN Y., Dislocation-density related constitutive modelling, in: “Unified constitutive
laws of plastic deformation”, (Eds. A.S. Krausz and K. Krausz), Academic Press 1996,
69–106.
[25] E STRIN Y., Dislocation-theory-based constitutive modelling: foundations and applications, J. Mater. Processing Tech. 80-81 (1998), 33–39.
[26] E STRIN Y., S LUYS L.J., B RECHET Y. AND M OLINARI A., A dislocation-based gradient
plasticity model, J. de Physique 8 (1998), 135–141.
[27] F RIED E., Continua described by a microstructural field, Z. Angew. Math. Phys. 47 (1996),
168–175.
[28] F RIED E. AND G URTIN M., Continuum theory of thermally-induced phase transitions
based on an order parameter, Physica D 68 (1993), 326–343.
[29] F RIED E. AND G URTIN M., Dynamic solid-solid transitions with phase characterized by
an order parameter, Physica D 72(1994), 287–308.
[30] F LECK N.A. AND H UTCHINSON J.W., A phenomenological theory for strain gradient
effects in plasticity, J. Mech. Phys. Solids 41 (1993), 1825–1857.
[31] F LECK N.A. AND H UTCHINSON J.W., Strain gradient plasticity, Adv. Appl. Mech. 33
(1997), 295–361.
[32] F OREST S., C AILLETAUD G. AND S IEVERT R., A Cosserat theory for elastoviscoplastic
single crystals at finite deformation, Arch. Mech. 49 (1997), 705–736.
[33] G URTIN M.E., On the thermodynamics of chemically reacting fluid mixtures, Arch. Rat.
Mech. Anal. 43 (1971), 198–212.
[34] G URTIN M.E., The nature of configurational forces, Arch. Rat. Mech. Anal. 131 (1995),
67–100.
234
B. Svendsen
[35] G URTIN M.E., On the plasticity of single crystals: free energy, microforces, plastic-strain
gradients, J. Mech. Phys. Solids 48 (2000), 989–1036.
[36] H ILL R. AND R ICE J.R., Constitutive analysis of elastic-plastic crystals at arbitrary
strain, J. Mech. Phys. Solids 20 (1972), 401–413.
[37] H UTCHINSON W., Elastic-plastic behaviour of polycrystalline metals and composites,
Proc. Roy. Soc. London 319 A (1976), 247–272.
[38] L E K.C. AND S TUMPF H., Non-linear continuum theory of dislocations Int. J. Eng. Sci.
34 (1996), 339–358.
[39] KOCKS U.F., Laws for work-hardening and low-temperature creep, J. Eng. Mat. Techn.
98 (1976), 76–85.
[40] KOCKS U.F., Constitutive behaviour based on crystal plasticity, in: “Constitutive equations for creep and plasticity” (Ed. A.K. Miller), Elsevier 1987, 1–88.
[41] K UHLMANN -W ILSDORF D., Theory of plastic deformation: properties of low-energy dislocation structures, Mat. Sci. Eng. 113 A (1989), 1.
[42] K R ÖNER E., Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen,
Arch. Rat. Mech. Anal. 4 (1960), 273–334.
[43] M AUGIN G., The method of virtual power in continuum mechanics; application to coupled
fields, Acta Mech. 35 (1980), 1–70.
[44] M AUGIN G., On internal variables and dissipative structures, Int. J. Non-Equil. Thermod.
15 (1990), 173–192.
[45] M AUGIN G. AND E PSTEIN M., Geometrical material structure of elastoplasticity, Int. J.
Plast. 14 (1998), 109–115.
[46] M AUGIN G. AND M USCHIK W., Thermodynamics with internal variables. I. General concepts, Int. J. Non-Equil. Thermod. 19 (1994), 217–249.
[47] M EIXNER J. AND R EIK H.G., Thermodynamik der irreversiblen Prozesse, in: “Handbuch
der Physik III/2”, Springer 1959.
[48] M ENZEL A. AND S TEINMANN P., On the continuum formulation of higher gradient plasticity for single and polycrystals, J. Mech. Phys. Solids 48 (2000), 1777–1796.
[49] M URA T., Micromechanics of defects in solids, Kluwer Publishers 1987.
[50] M ÜLLER I., A thermodynamic theory of mixtures of fluids, Arch. Rat. Mech. Anal. 28
(1968), 1–39.
[51] NAGHDI P. AND S RINIVASA A.R., A dynamical theory of structured solids. I. Basic developments, Phil. Trans. Roy. Soc. 345 A (1993), 424–458.
[52] NAGHDI P. AND S RINIVASA A.R., Characterisation of dislocations and their influence on
plastic deformation in single crystals, Int. J. Solids Structs. 7 (1994), 1157–1182.
[53] N OLL W., Materially uniform simple bodies with inhomogeneities, Arch. Rat. Mech. Anal.
27 (1967), 1–32.
[54] N YE J.F., Some geometric relations in dislocated crystals, Acta Metall. 1 (1953), 153–162.
[55] O RTIZ M. AND R EPETTO E.A., Nonconvex energy minimization and dislocation structures in ductile single crystals, J. Mech. Phys. Solids 47 (1999), 397–462.
[56] O RTIZ M., R EPETTO E.A. AND S TAINIER , L., A theory of dislocation structures, J. Mech.
Phys. Solids 48 (2000), 2077–2114.
Non-local continuum thermodynamic
235
[57] S ILHAV Ý M., The mechanics and thermodynamics of continuous media, Springer Verlag
1997.
[58] S HIZAWA K. AND Z BIB H.M., A thermodynamical theory of gradient elastoplasticity with
dislocation density tensor. I. Fundamentals, Int. J. Plast. 15 (1999), 899–938.
[59] S HU J.Y. AND F LECK N.A., Strain gradient crystal plasticity: size-dependent deformation
of bicrystals, J. Mech. Phys. Solids 47 (1999), 297–324.
[60] S HU J.Y., F LECK N.A., VAN DER G IESSEN E. AND N EEDLEMAN A., Boundary layers
in constrained plastic flow: comparison of nonlocal and discrete dislocation plasticity, J.
Mech. Phys. Solids 49, (2001), 1361–1395.
[61] S LUYS L.J. AND E STRIN Y., The analysis of shear banding with a dislocation-based
gradient plasticity model, Int. J. Solids Struct. 37 (2000), 7127–7142.
[62] S TEINMANN P., Views on multiplicative elastoplasticity and the continuum theory of dislocations, Int. J. Eng. Sci. 34 (1996), 1717–1735.
[63] S VENDSEN B., A thermodynamic formulation of finite-deformation elastoplasticity with
hardening based on the concept of material isomorphism, Int. J. Plast. 14 (1998), 473–488.
[64] S VENDSEN B., On the thermodynamics of isotropic thermoelastic materials with scalar
internal degrees of freedom, Cont. Mech. Thermodyn. 11 (1999), 247–262.
[65] S VENDSEN B., On the formulation of balance relations and configurational fields for materials with microstructure via invariance, Int. J. Solids Struct. 38 (2001), 700–740.
[66] S VENDSEN B., On the modeling of anisotropic elastic and inelastic material behaviour at
large deformation, Int. J. Solids Struct. (2000), in press.
[67] S VENDSEN B., A note on the concepts of material isomorphism and material uniformity
for inelastic gradient materials, in preparation.
[68] S VENDSEN B. AND R EESE S., Continuum thermodynamic modeling and simulation of
additional hardening due to deformation incompatibility; in: “Computational mechanics
of solid materials at large strains”, Proceedings of the IUTAM Symposium, August 20-24,
Stuttgart, Germany, 2001.
[69] T EODOSIU C., Dislocation modeling of crystalline plasticity; in: “Large plastic deformation of crystalline aggregates” (Ed. C. Teodosiu), CISM 376, Springer 1997, 21–80.
[70] T EODOSIU C. AND S IDEROFF F., A physical theory of finite elasto-viscoplastic behaviour
of single crystals, Int. J. Engrg. Sci. 14 (1976), 165–176.
[71] VALANIS K., A gradient theory of internal variables, Acta Mech. 116 (1996), 1–14.
[72] VALANIS K, A gradient thermodynamic theory of self-organization, Acta Mech. 127
(1998), 1–23.
[73] VAN DER G IESSEN E. AND N EEDLEMAN A., Discrete dislocation plasticity: a simple
planar model, Model. Simul. Mater. Sci. Eng. 3 (1995), 689–735.
[74] WANG C.C. AND B LOOM J., Material uniformity and inhomogeneity in anelastic bodies,
Arch. Rat. Mech. Anal. 53 (1974), 246–276.
236
AMS Subject Classification: 74A99, 74C20.
Bob SVENDSEN
Department of Mechanical Engineering
University of Dortmund
D-44227 Dortmund, GERMANY
e-mail: B. Svendsen